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$$

u_{tt} + 3u_t = u_{xx}\Rightarrow \varphi\psi'' + 3\varphi\psi' = \varphi''\psi.

$$

$$

u(0,t) = u(\pi,t) = 0

$$

$$

u(x,0) = 0\quad\text{and}\quad u_t(x,0) = 10

$$

\[\varphi(x) = A\cos kx + B\sin kx\\\]

\begin{alignat*}{3}

\psi(t) & = & C\exp\left(-\frac{3t}{2}\right)\exp\left[t\frac{\sqrt{9 - 4n^2}}{2}\right] + D\exp\left(-\frac{3t}{2}\right)\exp\left[-t\frac{\sqrt{9 - 4n^2}}{2}\right]

\end{alignat*}

The general sol would be

\begin{eqnarray}

u(x,t)&=&\exp\left[-\frac{3t}{2}\right]\sin x\left[A_1\cosh\frac{t\sqrt{5}}{2} + B_1\sinh\frac{t\sqrt{5}}{2}\right]\\

&+&\exp\left[-\frac{3t}{2}\right]\sum_{n = 2}^{\infty}\sin nx\left[C_n\cos t\frac{\sqrt{4n^2 - 9}}{2} + D_n\sin t\frac{\sqrt{4n^2 - 9}}{2}\right]

\end{eqnarray}

Correct?

u_{tt} + 3u_t = u_{xx}\Rightarrow \varphi\psi'' + 3\varphi\psi' = \varphi''\psi.

$$

$$

u(0,t) = u(\pi,t) = 0

$$

$$

u(x,0) = 0\quad\text{and}\quad u_t(x,0) = 10

$$

\[\varphi(x) = A\cos kx + B\sin kx\\\]

\begin{alignat*}{3}

\psi(t) & = & C\exp\left(-\frac{3t}{2}\right)\exp\left[t\frac{\sqrt{9 - 4n^2}}{2}\right] + D\exp\left(-\frac{3t}{2}\right)\exp\left[-t\frac{\sqrt{9 - 4n^2}}{2}\right]

\end{alignat*}

The general sol would be

\begin{eqnarray}

u(x,t)&=&\exp\left[-\frac{3t}{2}\right]\sin x\left[A_1\cosh\frac{t\sqrt{5}}{2} + B_1\sinh\frac{t\sqrt{5}}{2}\right]\\

&+&\exp\left[-\frac{3t}{2}\right]\sum_{n = 2}^{\infty}\sin nx\left[C_n\cos t\frac{\sqrt{4n^2 - 9}}{2} + D_n\sin t\frac{\sqrt{4n^2 - 9}}{2}\right]

\end{eqnarray}

Correct?

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